algebraic structure

TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' of finite
arity Arity () is the number of arguments In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logo ...
(typically
binary operation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s), and a finite set of identities, known as
axiom An axiom, postulate, or assumption is a Statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek () 'that which is thought worthy or ...
s, that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
involves a second structure called a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
, and an operation called ''scalar multiplication'' between elements of the field (called ''scalars''), and elements of the vector space (called ''vectors''). In the context of
universal algebraUniversal algebra (sometimes called general algebra) is the field of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geomet ...
, the set ''A'' with this
structure A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A ...
is called an ''algebra'', while, in other contexts, it is (somewhat ambiguously) called an ''algebraic structure'', the term ''algebra'' being reserved for specific algebraic structures that are
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
or
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a syst ...
over a
commutative ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies ...
. The properties of specific algebraic structures are studied in
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
. The general theory of algebraic structures has been formalized in universal algebra. The language of
category theory Category theory formalizes and its concepts in terms of a called a ', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or s). A has two basic properties: the ability to the arrows , and the exi ...
is used to express and study relationships between different classes of algebraic and non-algebraic objects. This is because it is sometimes possible to find strong connections between some classes of objects, sometimes of different kinds. For example,
Galois theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
establishes a connection between certain fields and groups: two algebraic structures of different kinds.

# Introduction

Addition and multiplication of real numbers are the prototypical examples of operations that combine two elements of a set to produce a third element of the set. These operations obey several algebraic laws. For example, ''a'' + (''b'' + ''c'') = (''a'' + ''b'') + ''c'' and ''a''(''bc'') = (''ab'')''c'' as the ''associative laws''. Also ''a'' + ''b'' = ''b'' + ''a'' and ''ab'' = ''ba'' as the ''commutative laws.'' Many systems studied by mathematicians have operations that obey some, but not necessarily all, of the laws of ordinary arithmetic. For example, rotations of an object in three-dimensional space can be combined by, for example, performing the first rotation on the object and then applying the second rotation on it in its new orientation made by the previous rotation. Rotation as an operation obeys the associative law, but can fail to satisfy the commutative law. Mathematicians give names to sets with one or more operations that obey a particular collection of laws, and study them in the abstract as algebraic structures. When a new problem can be shown to follow the laws of one of these algebraic structures, all the work that has been done on that category in the past can be applied to the new problem. In full generality, algebraic structures may involve an arbitrary collection of operations, including operations that combine more than two elements (higher
arity Arity () is the number of arguments In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logo ...
operations) and operations that take only one
argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
(
unary operation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s). The examples used here are by no means a complete list, but they are meant to be a representative list and include the most common structures. Longer lists of algebraic structures may be found in the external links and within '' :Algebraic structures.'' Structures are listed in approximate order of increasing complexity.

# Examples

## One set with operations

Simple structures: no
binary operation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
: * Set: a degenerate algebraic structure ''S'' having no operations. *
Pointed set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
: ''S'' has one or more distinguished elements, often 0, 1, or both. * Unary system: ''S'' and a single
unary operation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
over ''S''. * : a unary system with ''S'' a pointed set. Group-like structures: one binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition) as is done for ordinary multiplication of real numbers. * Magma or groupoid: ''S'' and a single binary operation over ''S''. *
Semigroup In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation. The binary operation of a semigroup is most often denoted multiplication, multiplicatively: ''x''·''y'', o ...
: an
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
magma. *
Monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...
: a semigroup with
identity element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. *
Group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
: a monoid with a unary operation (inverse), giving rise to
inverse element In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s. *
Abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
: a group whose binary operation is
commutative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. *
SemilatticeIn mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (mathematics), join (a least upper bound) for any nonempty set, nonempty finite set, finite subset. Duality (order theory), Dually, a meet-semilattic ...
: a semigroup whose operation is
idempotent Idempotence (, ) is the property of certain operations in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
and commutative. The binary operation can be called either meet or join. *
Quasigroup In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
: a magma obeying the Latin square property. A quasigroup may also be represented using three binary operations. * Loop: a quasigroup with
identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
. Ring-like structures or Ringoids: two binary operations, often called
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary Operation (mathematics), mathematical operations of arithmetic, with the ...

, with multiplication distributing over addition. *
Semiring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
: a ringoid such that ''S'' is a monoid under each operation. Addition is typically assumed to be commutative and associative, and the monoid product is assumed to distribute over the addition on both sides, and the additive identity 0 is an
absorbing elementIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
in the sense that 0 ''x'' = 0 for all ''x''. *
Near-ringIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
: a semiring whose additive monoid is a (not necessarily abelian) group. * Ring: a semiring whose additive monoid is an abelian group. *
Lie ring In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
: a ringoid whose additive monoid is an abelian group, but whose multiplicative operation satisfies the
Jacobi identity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
rather than associativity. *
Commutative ring In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...
: a ring in which the multiplication operation is commutative. *
Boolean ringIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
: a commutative ring with idempotent multiplication operation. *
Field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
: a commutative ring which contains a multiplicative inverse for every nonzero element. *
Kleene algebra In mathematics, a Kleene algebra ( ; named after Stephen Cole Kleene) is an idempotent (and thus partially ordered) semiring endowed with a closure operator. It generalizes the operations known from regular expressions. Definition Various inequ ...
s: a semiring with idempotent addition and a unary operation, the
Kleene star In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...
, satisfying additional properties. * *-algebra: a ring with an additional unary operation (*) satisfying additional properties. Lattice structures: two or more binary operations, including operations called meet and join, connected by the absorption law.Ringoids and Lattice (order), lattices can be clearly distinguished despite both having two defining binary operations. In the case of ringoids, the two operations are linked by the distributive law; in the case of lattices, they are linked by the absorption law. Ringoids also tend to have numerical model theory, models, while lattices tend to have set theory, set-theoretic models. * Complete lattice: a lattice in which arbitrary meet and joins exist. * Bounded lattice: a lattice with a greatest element and least element. * Complemented lattice: a bounded lattice with a unary operation, complementation, denoted by reverse Polish notation, postfix . The join of an element with its complement is the greatest element, and the meet of the two elements is the least element. * Modular lattice: a lattice whose elements satisfy the additional ''modular identity''. * Distributive lattice: a lattice in which each of meet and join distributive lattice, distributes over the other. Distributive lattices are modular, but the converse does not hold. * Boolean algebra (structure), Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation. This can be shown to be equivalent with the ring-like structure of the same name above. * Heyting algebra: a bounded distributive lattice with an added binary operation, relative pseudo-complement, denoted by infix →, and governed by the axioms ''x'' → ''x'' = 1, ''x'' (''x'' → ''y'') = ''x y'', ''y'' (''x'' → ''y'') = ''y'', ''x'' → (''y z'') = (''x'' → ''y'') (''x'' → ''z''). Arithmetics: two
binary operation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s, addition and multiplication. ''S'' is an infinite set. Arithmetics are pointed unary systems, whose
unary operation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is injective successor function, successor, and with distinguished element 0. * Robinson arithmetic. Addition and multiplication are Recursion, recursively defined by means of successor. 0 is the identity element for addition, and annihilates multiplication. Robinson arithmetic is listed here even though it is a variety, because of its closeness to Peano arithmetic. * Peano arithmetic. Robinson arithmetic with an axiom schema of mathematical induction, induction. Most ring and field axioms bearing on the properties of addition and multiplication are theorems of Peano arithmetic or of proper extensions thereof.

## Two sets with operations

module (mathematics), Module-like structures: composite systems involving two sets and employing at least two binary operations. * Group with operators: a group ''G'' with a set Ω and a binary operation Ω × ''G'' → ''G'' satisfying certain axioms. * module (mathematics), Module: an abelian group ''M'' and a ring ''R'' acting as operators on ''M''. The members of ''R'' are sometimes called scalar (mathematics), scalars, and the binary operation of ''scalar multiplication'' is a function ''R'' × ''M'' → ''M'', which satisfies several axioms. Counting the ring operations these systems have at least three operations. * Vector space: a module where the ring ''R'' is a division ring or
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
. * Graded vector space: a vector space with a Direct sum of modules, direct sum decomposition breaking the space into "grades". * Quadratic space: a vector space ''V'' over a field ''F'' with a quadratic form on ''V'' taking values in ''F''. algebra (ring theory), Algebra-like structures: composite system defined over two sets, a ring ''R'' and an ''R''-module ''M'' equipped with an operation called multiplication. This can be viewed as a system with five binary operations: two operations on ''R'', two on ''M'' and one involving both ''R'' and ''M''. * Algebra over a ring (also ''R-algebra''): a module over a
commutative ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies ...
''R'', which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and Bilinear map, linearity with respect to multiplication by elements of ''R''. The theory of an algebra over a field is especially well developed. * Associative algebra: an algebra over a ring such that the multiplication is associative property, associative. * Nonassociative algebra: a module over a commutative ring, equipped with a ring multiplication operation that is not necessarily associative. Often associativity is replaced with a different identity, such as alternative algebra, alternation, the
Jacobi identity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, or the Jordan identity. * Coalgebra: a vector space with a "comultiplication" defined dually to that of associative algebras. * Lie algebra: a special type of nonassociative algebra whose product satisfies the
Jacobi identity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. * Lie coalgebra: a vector space with a "comultiplication" defined dually to that of Lie algebras. * Graded algebra: a graded vector space with an algebra structure compatible with the grading. The idea is that if the grades of two elements ''a'' and ''b'' are known, then the grade of ''ab'' is known, and so the location of the product ''ab'' is determined in the decomposition. * Inner product space: an ''F'' vector space ''V'' with a definite bilinear form . Four or more binary operations: * Bialgebra: an associative algebra with a compatible coalgebra structure. * Lie bialgebra: a Lie algebra with a compatible bialgebra structure. * Hopf algebra: a bialgebra with a connection axiom (antipode). * Clifford algebra: a graded associative algebra equipped with an exterior product from which may be derived several possible inner products. Exterior algebras and geometric algebras are special cases of this construction.

# Hybrid structures

Algebraic structures can also coexist with added structure of non-algebraic nature, such as Partially ordered set#Formal definition, partial order or a topology. The added structure must be compatible, in some sense, with the algebraic structure. * Topological group: a group with a topology compatible with the group operation. * Lie group: a topological group with a compatible smooth manifold structure. * Ordered groups, ordered rings and ordered fields: each type of structure with a compatible partial order. * Archimedean group: a linearly ordered group for which the Archimedean property holds. * Topological vector space: a vector space whose ''M'' has a compatible topology. * Normed vector space: a vector space with a compatible norm (mathematics), norm. If such a space is complete metric space, complete (as a metric space) then it is called a Banach space. * Hilbert space: an inner product space over the real or complex numbers whose inner product gives rise to a Banach space structure. * Vertex operator algebra * Von Neumann algebra: a *-algebra of operators on a Hilbert space equipped with the weak operator topology.

# Universal algebra

Algebraic structures are defined through different configurations of axioms. Universal algebra abstractly studies such objects. One major dichotomy is between structures that are axiomatized entirely by ''identities'' and structures that are not. If all axioms defining a class of algebras are identities, then this class is a variety (universal algebra), variety (not to be confused with algebraic varieties of algebraic geometry). Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universal quantifier, universally quantified over the relevant universe (mathematics), universe. Identities contain no Logical connective, connectives, Quantification (science), existentially quantified variables, or finitary relation, relations of any kind other than the allowed operations. The study of varieties is an important part of
universal algebraUniversal algebra (sometimes called general algebra) is the field of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geomet ...
. An algebraic structure in a variety may be understood as the quotient (universal algebra), quotient algebra of term algebra (also called "absolutely free object, free algebra") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given signature (logic), signatures generate a free algebra, the term algebra ''T''. Given a set of equational identities (the axioms), one may consider their symmetric, transitive closure ''E''. The quotient algebra ''T''/''E'' is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator ''m'', taking two arguments, and the inverse operator ''i'', taking one argument, and the identity element ''e'', a constant, which may be considered an operator that takes zero arguments. Given a (countable) set of variables ''x'', ''y'', ''z'', etc. the term algebra is the collection of all possible term (logic), terms involving ''m'', ''i'', ''e'' and the variables; so for example, ''m''(''i''(''x''), ''m''(''x'', ''m''(''y'',''e''))) would be an element of the term algebra. One of the axioms defining a group is the identity ''m''(''x'', ''i''(''x'')) = ''e''; another is ''m''(''x'',''e'') = ''x''. The axioms can be represented a
trees
These equations induce equivalence classes on the free algebra; the quotient algebra then has the algebraic structure of a group. Some structures do not form varieties, because either: # It is necessary that 0 ≠ 1, 0 being the additive
identity element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
and 1 being a multiplicative identity element, but this is a nonidentity; # Structures such as fields have some axioms that hold only for nonzero members of ''S''. For an algebraic structure to be a variety, its operations must be defined for ''all'' members of ''S''; there can be no partial operations. Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., Field (mathematics), fields and division rings. Structures with nonidentities present challenges varieties do not. For example, the direct product of two
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
s is not a field, because $\left(1,0\right)\cdot\left(0,1\right)=\left(0,0\right)$, but fields do not have zero divisors.

# Category theory

Category theory is another tool for studying algebraic structures (see, for example, Mac Lane 1998). A category is a collection of ''objects'' with associated ''morphisms.'' Every algebraic structure has its own notion of homomorphism, namely any function (mathematics), function compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a category (mathematics), category. For example, the category of groups has all Group (mathematics), groups as objects and all group homomorphisms as morphisms. This concrete category may be seen as a category of sets with added category-theoretic structure. Likewise, the category of topological groups (whose morphisms are the continuous group homomorphisms) is a category of topological spaces with extra structure. A forgetful functor between categories of algebraic structures "forgets" a part of a structure. There are various concepts in category theory that try to capture the algebraic character of a context, for instance * algebraic category * essentially algebraic category * presentable category * locally presentable category * Monad (category theory), monadic functors and categories * universal property.

# Different meanings of "structure"

In a slight abuse of notation, the word "structure" can also refer to just the operations on a structure, instead of the underlying set itself. For example, the sentence, "We have defined a ring ''structure'' on the set $A$," means that we have defined ring (mathematics), ring ''operations'' on the set $A$. For another example, the group $\left(\mathbb Z, +\right)$ can be seen as a set $\mathbb Z$ that is equipped with an ''algebraic structure,'' namely the ''operation'' $+$.

* Free object * List of algebraic structures * Mathematical structure * Outline of algebraic structures * Signature (logic) * Structure (mathematical logic)

# References

* * * ; Category theory * *